SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\begin{array}{l} \mathcal {L} ^ {(F)} (\theta) = \mathcal {L} ^ {(F, 1)} (\theta) + \mathcal {L} ^ {(F, 2)} (\theta) \\ = \mathbb {E} _ {\substack {\boldsymbol {x} = g _ {\theta} (\boldsymbol {z}) \\ \boldsymbol {z} \sim p _ {z}}} \frac {1}{2} \left[ \| \boldsymbol {s} _ {q} (\boldsymbol {x}) \| _ {2} ^ {2} - \| \boldsymbol {s} _ {\theta^ {\dagger}} (\boldsymbol {x}) \| _ {2} ^ {2} + 2 \nabla_ {\boldsymbol {x}} \left[ \boldsymbol {s} _ {q} (\boldsymbol {x}) - \boldsymbol {s} _ {\theta^ {\dagger}} (\boldsymbol {x}) \right] \right]. \tag{9} \\ \end{array}
$$
And we formally define the gradient operator
$$
\operatorname {G r a d} ^ {(F)} (\theta) := \frac {\partial}{\partial \theta} \mathcal {L} ^ {(F)} (\theta). \tag {10}
$$
Similar to the KL divergence case as discussed in Section 3.1, our overall training approach of such a Fisher divergence minimization consists of two phases: the score-estimation phase, and the Fisher divergence minimization phase. The score-estimation phase is the same as the one used for KL training, while the Fisher divergence minimization updates the implicit sampler by minimizing objective equation 9. Since the introduced method aims to minimize the Fisher divergence, we name our training the Fisher training for an implicit sampler.
Architecture choice of score network. Notice that for Fisher training, the objective equation 9 needs the calculation of the gradient of the scoring network  $\nabla_{\pmb{x}} s_{\phi}(\pmb{x})$ , so it requires the scoring network to be point-wisely differentiable. Commonly used activation functions such as ReLU or LeakyReLU do not satisfy the differentiability property. In our further experiments, we use differentiable activation functions for Fisher training.
We formally give a unified Algorithm for the training of implicit samplers with KL, Fisher, and Combine training in Algorithm 1. We use the standard score-matching objective as an example of score estimation phases, other score estimation methods are also suitable. The notation  $s_{\phi}^{(d)}$  and  $x_{d}$  represent the  $d$ -th component of the score network  $s_{\phi}(.)$  and the data  $x$ , and  $D$  means the data dimension. The notation  $\mathrm{sg}(.)$  represents stopping the parameter dependence of the input.
Algorithm 1: Training Algorithm for Implicit Samplers
Input: un-normalized target log  $q(x)$  , latent distribution  $p_z(z)$  , implicit sampler  $g_{\theta}(.)$  , score network  $\boldsymbol {s}_{\phi}(.)$  , mini-batch size B, max iteration M.   
Randomly initialize  $(\theta^{(0)},\phi^{(0)})$    
for t in 0:M do   
// update score network parameter   
Get mini-batch without parameter dependence   
 $x_{i} = \mathrm{sg}[g_{\theta (t)}(\boldsymbol {z}_{i})],\boldsymbol {z}_{i}\sim p_{z}(\boldsymbol {z}),i = 1,\dots,B.$    
Calculate score matching or related objective:   
 $\mathcal{L}(\phi) = \frac{1}{B}\sum_{i = 1}^{B}\left[\| \boldsymbol{s}_{\phi}(\boldsymbol{x}_i)\| _2^2 +2\sum_{d = 1}^{D}\frac{\partial\boldsymbol{s}_{\phi}^{(d)}(\boldsymbol{x}_i)}{\partial\boldsymbol{x}_d}\right].$  Minimize  $\mathcal{L}_{SM}(\phi)$  to get  $\phi^{(t + 1)}$    
// update sampler parameter   
Get mini-batch samples with parameter-dependecne   
 $\boldsymbol {x}_i = g_{\theta (t)}(\boldsymbol {z}_i),\boldsymbol {z}_i\sim p_z(\boldsymbol {z}),i = 1,\dots,B.$    
Calculate Gradient grad with formula 3 or 10.   
Update parameter  $\theta$  with grad and gradient-based optimization algorithms to get  $\theta^{(t + 1)}$    
end   
return  $(\theta ,\phi)$
3.3 Connection to Fisher-Stein's Sampler
In this section, we provide an analysis of the connection between our proposed Fisher training method and Fisher-Stein's sampler proposed in Hu et al. [11]. Surprisingly, we discover that the training method used in Fisher-Stein's sampler is implicitly equivalent to our proposed Fisher training. To establish this connection, we adopt the notations introduced in the introduction of Fisher-Stein's sampler in Section 2.1. We summarize our findings in a theorem, and for a comprehensive understanding of the proof and analysis, we provide detailed explanations in Appendix A.5.
Proposition 1. Estimating the sampler's score function  $s_{\phi}(.)$  with score matching is equivalent to the maximization of test function  $\mathbf{f}$  of Fisher-Stein's Discrepancy objective (15). More specially, the optimal score estimation  $S^{*}$  and Fisher-Stein's optimal test function  $\mathbf{f}^{*}$  satisfy
$$
\boldsymbol {f} ^ {*} (\boldsymbol {x}) = \frac {1}{2 \lambda} \left[ \nabla_ {\boldsymbol {x}} \log q (\boldsymbol {x}) - \boldsymbol {s} ^ {*} (\boldsymbol {x}) \right].
$$
With Proposition 2, we can prove a Theorem that states that the Fisher-Stein training coincides with our proposed Fisher training.
Theorem 1. Assume  $\pmb{f}^{*}$  is the optimal test function that maximizes Fisher-Stein's objective 15. Then the minimization part of objective 15 for sampler  $G_{\theta}$  with the gradient-based algorithm is equivalent to the Fisher training, i.e. the minimization objective of 15 shares the same gradient with  $\mathrm{Grad}^{(F)}(\theta)$ .
Theorem 1 states that Fisher-Stein's sampler and its training method are equivalent to our proposed Fisher training. The two methods differ only in the parametrization of the score network (and the test function  $\pmb{f}$ ) and the motivation that they are proposed. It is one of our contributions to point out
this equivalence. Our analysis reveals the underlying connection between the two training methods, shedding light on the similarities and shared principles between them. By establishing this connection, we provide a deeper understanding of the theoretical foundations of our proposed Fisher training approach and its relationship to existing techniques in the field.
In previous sections, we have established the KL and Fisher training methods for implicit samplers. Both methods only require access to an un-normalized potential function of the target distribution. In this section, we aim to apply our introduced training methods on three sampling benchmarks whose scales vary across both low (2-dimensional targets) to high (784-dimensions image targets). We compare both the implicit sampler and our training methods to other competitor methods and samplers. All experiments demonstrate the significant advantage of the implicit sampler and the proposed training methods on both the sampling efficiency and the sample quality.
4.1 2D Synthetic Target Sampling
Sample quality on 2D targets. In this section, we refer to the open-source implementation of Sharrock and Nemeth [49] and train samplers on eight 2D target distributions. We compare our neural samplers with 3 MCMC baselines: Stein variational gradient descent (SVGD) [50], Langevin dynamics (LD) [51], and Hamiltonian Monte Carlo (HMC) [52]; 1 explicit baseline: coupling normalizing flow [53]; and 2 implicit samplers: KSD neural sampler (KSD-NS) [11] and SteinGan [54]. All implicit samplers have the same neural architectures, i.e., four layer MLP with 400 hidden units at each layer and ELU activation functions for sampler and score network (if necessary).
The kernelized Stein's discrepancy (KSD) [55] is a popular metric for evaluating the sample quality of Monte Carlo samplers [55, 40]. We evaluate the KSD with IMQ kernel (implemented by an open-source package, the sgmcmcjax<sup>5</sup>) on all target distributions as the metric reported in Table 1.
Table 1: KSD Comparison of Samplers. If not especially emphasized, we set the Stepsize=0.01, iterations=500, num particles=500, num chains=20.
SAMPLER	GAUSSIAN	MOG2	ROSEN BROCK	DONUT	FUNNEL	SQUIGGLE
MCMC
SVGD(500)	0.013 ± 0.001	0.044 ± 0.006	0.053 ± 0.002	0.057 ± 0.004	0.052 ± 0.001	0.024 ± 0.002
LD(500)	0.107 ± 0.025	0.099 ± 0.008	0.152 ± 0.030	0.107 ± 0.020	0.116 ± 0.029	0.139 ± 0.030
HMC(500)	0.094 ± 0.020	0.106 ± 0.020	0.134 ± 0.034	0.113 ± 0.020	0.135 ± 0.010	0.135 ± 0.033
NEURAL
COUP-FLOW	0.102 ± 0.028	0.158 ± 0.019	0.150 ± 0.026	0.239 ± 0.013	0.269 ± 0.019	0.130 ± 0.026
KSD-NS	0.206 ± 0.043	1.129 ± 0.197	1.531 ± 0.058	0.341 ± 0.039	0.396 ± 0.221	0.462 ± 0.065
STEINGAN	0.091 ± 0.013	0.131 ± 0.011	0.121 ± 0.022	0.104 ± 0.013	0.129 ± 0.020	0.124 ± 0.018
FISHER-NS	0.095 ± 0.016	0.118 ± 0.013	0.157 ± 0.030	0.179 ± 0.028	7.837 ± 1.614	0.202 ± 0.037
KL-NS	0.099 ± 0.015	0.104 ± 0.015	0.123 ± 0.021	0.109 ± 0.015	0.115 ± 0.012	0.118 ± 0.024
Settings. For all MCMC samplers, we set the number of iterations to 500, which we find is enough for convergence. For SVGD and LD, we set the sampling step size to 0.01. For the HMC sampler, we optimize and find the step size to be 0.1, and LeapFrog updates to 10 work the best. For Coupling Flow, we follow Dinh et al. [53], and use 3 invertible blocks, with each block containing a 4-layer MLP with 200 hidden units and Gaussian Error Linear Units (GELU) activations. The total parameters of flow are significantly larger than neural samplers. We find that adding more coupling blocks does not lead to better performance. For all targets, we train each neural sampler with the Adam optimizer with the same learning rate of 2e-5 and default bete. We use the same batch size of 5000 for 10k iterations when training all neural samplers. We evaluate the KSD for every 200 iterations with 500 samples with 20 repeats for each time. We pick the lowest mean KSD among 10k training iterations as our final results. Since our proposed Fisher neural sampler and KL neural sampler require learning the score network, we find that using 5-step updates of the score network for each update of the neural sampler works well.
<sup>4</sup>https://github.com/luissharrock/coin-svgd
<sup>5</sup>https://github.com/jeremiecouillon/SGMCMCJax
Performance. Table 1 shows the numerical comparison between the mentioned samplers. The SVGD performs significantly the best among all samplers. However, the SVGD has a more heavy computational cost when the number of particles grows because its update requires matrix computation for a large matrix. The LD and HMC perform almost the same. The KL-NS performs the best across almost all targets, slightly better than LD and HMC on each target. The SteinGAN performs second and is closely comparable to KL-NS. In theory, both the KL-NS and the SteinGAN aim to minimize the KL divergence between the sampler and the target distribution in different ways, so we are not surprised by their similar performances. The Coupling Flow performs overall third, but it fails to correctly capture the Rosenbrock target. We believe more powerful flows, such as stochastic variants or flows with more complex blocks will lead to better performance, but these enhancements will inevitably bring in more computational complexity. The Fisher-NS performs the fourth, with 1 failure case on the Funnel target. We find that the KSD-NS is hard to tune in practice, which has two failure cases. Besides, the KSD-NS has a relatively high computational cost because it requires differentiation through the empirical KSD, which needs large matrix computation when the batch size is large. Overall, the one-shot KL-NS has shown strong performance, outperforming LD and HMC with multiple iterations. We also visualize the sample results on five distributions with hard-to-sample characteristics such as multi-modality and periodicity, as shown in Figure 2.
Comparison of Computational Costs. To demonstrate the significant advantage of our proposed implicit samplers over other samplers, we evaluate the efficiency of each sampler under the same environment to measure the wall-clock inference time. In below Table 2, we summarize the computational costs (wall-clock time) of SVGD, HMC, and LD together with our KL-NS neural sampler to compare the computational costs of each method. The eight targets are analytic targets while the EBM is a neural target. Each sampler can generate samples with comparable qualities (i.e. their KSD values are comparable). Table 2 records the wall-clock inference time (seconds) for each sampler when generating 1k samples.
Table 2: Inference Time Comparison of MCMC Samplers and Neural Samplers (seconds). The 2D experiment is conducted on an 8-CPU cluster with PyTorch of 1.8.1, while the EBM experiment is on 1 Nvidia Titan RTX GPU with PyTorch 1.8.1. Unless especially emphasizing, we set the stepsize=0.01, iterations=500, num particles=1000, num repeats=100.
SAMPLEER	GAUSSIAN	MOG2	ROSEN BROCK	DONUT	FUNNEL
SVGD(500)	26.4224 ± 0.6000	26.7897 ± 0.5712	26.1088 ± 0.4666	26.3632 ± 0.4290	26.0555 ± 0.4232
LD(500)	0.1035 ± 0.0003	0.2168 ± 0.0008	0.1284 ± 0.0002	0.1100 ± 0.0003	0.1339 ± 0.0032
HMC(500)	0.3438 ± 0.0017	1.7125 ± 0.0154	0.6725 ± 0.0044	0.3854 ± 0.0015	0.6837 ± 0.0018
KL-NS	0.0014 ± 0.0000	0.0014 ± 0.0000	0.0014 ± 0.0000	0.0014 ± 0.0000	0.0014 ± 0.0000
The LD and HMC with 500 iterations have comparable (slightly worse) performance than the Neural Sampler, however, their wall-clock time for obtaining 1k samples is significantly larger than Neural samplers. Let's take the simple mixture-of-2-gaussian (MOG2) target (with analytic potential and scores) as an example, the neural sampler is  $1.7125 / 0.0014 = 1223$  times faster than HMC and  $0.2168 / 0.0014 = 154$  times faster than LD. If one wants to keep the same inference time for LD and HMC to be comparable with the neural sampler, the LD will only have  $500 / 154 = 3.2$  iterations, while the HMC will only have  $500 / 1223 = 0.4$  iterations. It is impossible to obtain promising samples with only 3.2 or 0.4 MCMC iterations.
In conclusion, for both analytic and neural targets, the neural samplers show a significant efficiency advantage over MCMC samplers with better (comparable) performances. This shows that neural samplers have potentially wider use on many sampling tasks for which high efficiency and low computational costs are demanded.
4.2 Bayesian Regression
Test accuracy compared with Stein sampler. In the previous section, we validate the sampler on low-dimensional 2D target distributions. The Bayesian logistic regression provides a source of real-world target distributions with medium dimensions (10-100). In this section, we compare our sampler with the most related sampler, Stein's sampler that is proposed in Hu et al. [11]. We compare the test accuracy of our proposed sampler against Stein's samplers and MCMC samplers.
Sampler 1NFE
Ebm 50NFE
Ebm 100NFE
Ebm 150NFE
Ebm 200NFE
Figure 1: Samples from implicit samplers and energy-based models trained on MNIST datasets. Our trained sampler only requires one NFE, which is comparable to EBM samples with 100 NFEs.
Figure 2: Samples from implicit samplers trained with KL training method on five target distributions. Up: Visualization of target un-normalized density; Below: samples from our trained sampler.
Experiment settings. We follow the same setting as Hu et al. [11] on the Bayesian logistic regression tasks. The Covertype data set [11] has 54 features and 581,012 observations. It has been widely used as a benchmark for Bayesian inference. More precisely, following the Hu et al. [11], we assume prior of the weights is  $p(w|\alpha) = \mathcal{N}(w;0,\alpha^{-1})$  and  $p(\alpha) = \mathrm{Gamma}(\alpha ;1,0.01)$ . The data set is randomly split into the training set (80%) and the testing set (20%). Our goal is to train an implicit sampler that can sample from posterior distribution efficiently. The sampler trained with Fisher training is the same as the Fisher-Neural Sampler, which achieves the best test accuracy, as we have proved in previous sections. The sampler trained with the KL method gives the secondary test accuracy in Table 3. This experiment shows that our proposed training method is capable of handling real-world Bayesian inference with a medium dimension. After training, the implicit sampler can result in the best accuracy while achieving hundreds of times faster than MCMC methods. For detailed settings, please refer to Appendix B.
Table 3: Test Accuracies for Bayesian Logistic Regression on Covertype Dataset. The Fisher-NS is equivalent to a sampler with our proposed Fisher training, i.e. Fisher-IS.
SGLD
75.09% ± 0.20%	DSVI
73.46% ± 4.52%
SVGD
74.76% ± 0.47%	STEINGAN
75.37% ± 0.19%
FISHER-NS
76.22% ± 0.43%	KL-IS(Ours)
75.95% ± 0.002%
4.3 Sampling from Energy-based Models
A main advantage of an implicit sampler is its inference efficiency. This advantage benefits applications for which the default sampling method is inefficient, such as sampling from high-dimensional energy-based models. In this section, we use our proposed KL training method for an application of learning an efficient implicit sampler to sample from a pre-trained energy-based model.
Table 4: Comparison of Sampling Efficiency and Performance.
Model	NFE	Wall-clock Time	FLOPS	FID	KID
EBM	250	0.8455	0.58G×250	20.95	0.0097
EBM	200	0.6658	0.58G×200	20.92	0.0111
EBM	150	0.5115	0.58G×150	21.32	0.0169
EBM	100	0.3455	0.58G×100	30.35	0.0742
EBM	50	0.1692	0.58G×50	52.55	0.2061
KL Sampler (ours)	1	0.0012	1.11G	22.29	0.0045
Experiment settings. An energy-based model (EBM) [56] uses a neural network to parametrize a negative energy function, i.e. the logarithm of some un-normalized distribution. After training, obtaining samples from an EBM usually requires running an annealed Markov Chain Monte Carlo [4] which is computationally inefficient. In this section, we pre-train a deep EBM [57] on the MNIST dataset and apply our proposed KL training method for learning a neural implicit sampler to learn to sample from the pre-trained EBM efficiently. To take the inductive biases of image data, we parametrize our sampler via the stacking of several convolutional neural blocks. The neural sampler takes a random Gaussian input with a dimension of 128 and outputs a  $32 \times 32$  tensor which has the same size as the resolution on which the EBM has been trained. We visualize the samples drawn from EBM with  $50+$  evaluations of the forward and backward pass of EBMs with annealed Langevin dynamics and samples from our sampler which only requires a single forward pass of the sampler's neural network. We train an MNIST classifier and compute the Frechet Inception Distance (FID) [58] and the KID [59] as an evaluation metric.
Performance. As is shown in Table 4, our sampler generates indistinguishable samples from the EBM but is about  $100+$  more efficient than annealed Langevin dynamics. To demonstrate the sampling efficiency of our sampler, we compare the efficiency and computational costs of EBM and our sampler, by listing some major metrics on computational performance in Table 4. The EBM has a total amount of  $9.06M$  parameters and a base FLOPS of  $0.58G$ , while the sampler has  $12.59M$  parameters and a base FLOPS of  $1.11G$ . Generating samples from EBM requires running MCMC chains, which consist of up to 200 neural functional evaluations (NFEs) of the EBM. We compare the efficiency and performance of EBM with different NFEs and find that with less than 100 NFEs the EBM can not generate realistic samples, while our sampler only needs one single NFE for good samples. This high-dimensional EBM sampling experiment demonstrates the scalability of KL training methods and their potential for accelerating large-scale energy-based model sampling.
5 Limitations and Future Works
We have presented two novel approaches for training an implicit sampler to sampler from unnormalized density. We show in theory that our training methods are equivalent to minimizing the KL or the Fisher divergences. We systematically compare the proposed training methods against each other and against other samplers (NFs and MCMC). Besides, we conduct an experiment that trains a convolutional neural sampler to learn to sample from a pre-trained EBM. This experiment shows that our implicit sampler and corresponding training have great potential to improve the efficiency of modern machine learning models such as energy-based models or diffusion models.
However, there are still limitations to our proposed methods. First, the score estimation is not computationally cheap, so developing a more efficient training algorithm by eliminating the score estimation phase is an important research direction. Besides, now the sampler is only limited to sampling problems. How to extend the methodology for other applications such as generative modeling is another interesting direction.
This work has been supported by the National Natural Science Foundation of China (No. 12271011) and the Beijing Natural Science Foundation (Z190001).
<sup>6</sup>The Fisher training method fails when learning to sample from EBM. Besides, the Fisher training method requires backpropagating through models multiple times, therefore is computationally inefficient.
[59] M. Binkowski, D. J. Sutherland, M. Arbel, and A. Gretton, “Demystifying mmd gans,” in International Conference on Learning Representations, 2018.
[60] L. K. Wenliang and H. Kanagawa, "Blindness of score-based methods to isolated components and mixing proportions," arXiv preprint arXiv:2008.10087, 2020.
[61] W. Gong and Y. Li, "Interpreting diffusion score matching using normalizing flow," in ICML Workshop on Invertible Neural Networks, Normalizing Flows, and Explicit Likelihood Models, 2021.
[62] J. A. Blackard and D. J. Dean, “Comparative accuracies of artificial neural networks and discriminant analysis in predicting forest cover types from cartographic variables,” Computers and Electronics in Agriculture, vol. 24, pp. 131–151, 1999.
[63] M. K. Titsias and M. Lázaro-Gredilla, “Doubly stochastic variational bayes for non-conjugate inference,” in International Conference on Machine Learning, 2014.
[64] D. Hendrycks and K. Gimpel, “Gaussian error linear units (gelus),” arXiv: Learning, 2016.
[65] K. He, X. Zhang, S. Ren, and J. Sun, “Deep residual learning for image recognition,” 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 770–778, 2015.
[66] S. Zagoruyko and N. Komodakis, "Wide residual networks," arXiv preprint arXiv:1605.07146, 2016.
[67] M. F. Hutchinson, “A stochastic estimator of the trace of the influence matrix for laplacian smoothing splines,” Communications in Statistics - Simulation and Computation, vol. 18, pp. 1059-1076, 1989.
A.1 Detailed Derivations of Vanishing Term of KL Training Method
Recall the vanishing term
$$
\begin{array}{l} \mathbb {E} _ {\boldsymbol {z} \sim p _ {z}} \left[ \frac {\partial \log p _ {\theta} (\boldsymbol {x})}{\partial \theta} | _ {\boldsymbol {x} = g _ {\theta} (\boldsymbol {z})} \right] \\ = \mathbb {E} _ {\boldsymbol {x} \sim p _ {\theta}} \left[ \frac {\partial \log p _ {\theta} (\boldsymbol {x})}{\partial \theta} \right] = \int \frac {\partial p _ {\theta} (\boldsymbol {x})}{\partial \theta} d \boldsymbol {x} = \frac {\partial}{\partial \theta} \int p _ {\theta} (\boldsymbol {x}) d \boldsymbol {x} = \frac {\partial}{\partial \theta} 1 = \mathbf {0}. \\ \end{array}
$$
The equality holds if  $p_{\theta}(\pmb{x})$  satisfies the conditions (1).  $p_{\theta}(\pmb{x})$  is Lebesgue integrable for  $\pmb{x}$  with each  $\theta$ ; (2). For almost all  $\pmb{x} \in \mathbf{R}^D$ , the partial derivative  $\partial p_{\theta}(\pmb{x}) / \partial \theta$  exists for all  $\theta \in \Theta$ . (3) there exists an integrable function  $g(.) : \mathbf{R}^D \to \mathbf{R}$ , such that  $p_{\theta}(\pmb{x}) \leq g(\pmb{x})$  for all  $\pmb{x}$  in its domain. Then the derivative w.r.t  $\theta$  can be exchanged with the integral over  $\pmb{x}$ , i.e.
$$
\int \frac {\partial}{\partial \theta} p _ {\theta} (\boldsymbol {x}) \mathrm {d} \boldsymbol {x} = \frac {\partial}{\partial \theta} \int p _ {\theta} (\boldsymbol {x}) \mathrm {d} \boldsymbol {x}.
$$
A.2 Detailed Derivations of Fisher Training Methods
Recall the definition of the Fisher divergence,
$$
\mathcal {D} ^ {(F)} \left(p _ {\theta}, q\right) := \mathbb {E} _ {p _ {\theta}} \frac {1}{2} \| \nabla_ {\boldsymbol {x}} \log p _ {\theta} (\boldsymbol {x}) - \nabla_ {\boldsymbol {x}} \log q (\boldsymbol {x}) \| _ {2} ^ {2} = \mathbb {E} _ {\boldsymbol {z} \sim p _ {z}} \frac {1}{2} \| \boldsymbol {s} _ {\theta} \left(g _ {\theta} (\boldsymbol {z})\right) - \boldsymbol {s} _ {q} \left(g _ {\theta} (\boldsymbol {z})\right) \| _ {2} ^ {2}. \tag {11}
$$